1

Ibn Sina’s explanation ofreductio ad absurdum.

Wilfrid HodgesHerons Brook, Sticklepath, Okehampton

November 2011http://wilfridhodges.co.uk

2

ARISTOTLE

CHRYSIPPUS

IBN SINA

ABELARD LEIBNIZ

FREGE

MODERNS

Roman Empirecommentators

Latin linethroughBoethius

Greek linethroughAlexandrians

Scholastics

WESTERN LOGIC — THE BIG NAMES

Later Arabic logicians

3

Reductio ad absurdum (RAA):to prove φ from known facts ψ we argue

��

�

@@@

¬φ ψ

��

�

@@@⊥

4

How to justify using this derivation asa demonstration of φ given ψ?

Ibn Sına’s answer:Here as often, we mean more than we say.Common practice is to introduce the false assumption ¬φwith ‘if’, but not repeat ‘If ¬φ’ when we derive thingsfrom it, although in strict logic the condition is needed.

Restoring the implied condition gives a richer derivationas follows.

5

We restore the implied ‘If ¬φ then . . . ’ on ¬φ itself andevery formula that depends on it:

READ��

�

@@@

¬φ ψ

��

�

@@@⊥

THINK�

��

@@@

(¬φ→ ¬φ) ψ�

����

QQQQQ

(¬φ→⊥)

φ

NB the richer derivation is still valid,and (¬φ→ ¬φ) is a self-evident axiom.

6

Strictly Ibn Sına describes a more complicated derivation:

���

@@@

(¬φ→ ¬φ) ψ�

��

��

QQQQQ

(¬φ→ χ)

���

@@@¬χ

φ

7

Two probable reasons why he adds χ:

1. To express that the reasoner must not just deduce acontradiction, but also know that the contradiction isa contradiction.

2. To separate the ‘assumption’ part of RAA from the‘constructively dubious’ part.

2 interprets Aristotle. Ibn Sına thinks the ‘constructivelydubious’ part is a minor problem, and the challenge is tomake sense of an assumption made and later discharged.His explanation covers this, not just for RAA.

8

In what follows we consider:(I) what Ibn Sına counted as ‘justifying’ RAA;

(II) his evidence for the missing conditions;(III) his reason for thinking that the derivation remains

valid when the conditions are added.

9

(I) What counts as justifying RAA?

Today we require explanations to be precise andunambiguous.Any general principles invoked should be universallytrue.

Traditional justifications were more like answers to achild’s question ‘Why?’.Hard to find anything universally true about naturallanguage.

For Ibn Sına, medical reasoning was one paradigm,and all general principles in medicine are‘true in most cases’ (akt

¯arı).

10

For Ibn Sına and many traditional logicians,explaining a rule of reasoning means showing howsomeone using it could be rightly convinced of what itproves.

So the explanation should use only concepts andprinciples already available to people who use the rule.

Thus Ibn Sına rejects justifications that(1) are unreasonably complicated, or that(2) bring in metatheory or(3) use an artificial language.

11

“[One commentator makes various alterations andadditions which] lengthen the discussion but giveus no new information. [By contrast] the accountwe have given is exactly the RAA syllogism itself,no more and no less.” (Ibn Sına, Sifa’ viii.3)

This is an exaggeration, obviously.To repeat the original argument, no more and no less,would hardly be a justification of it.But Ibn Sına can claim that his justification uses no newconcepts and no added steps.

12

Ibn Sına also rejects justifications based on dialogue,as for example:

The form in which Euclid argues, supposes an op-ponent; and the whole argument then stands as fol-lows. “When X is Y, you grant that P is Q; but yougrant that P is not Q. I say that X is not Y. If you denythis you must affirm that X is Y, of which you admitit to be a consequence that P is Q. But you grant thatP is not Q; therefore” (etc. etc.) (De Morgan, Elementsof Trigonometry p. 5)

(This is a figment of De Morgan’s imagination.It draws on Aristotle, not on Euclid.)

13

Ibn Sına often refers to dialogues where we show ouropponent that she is wrong to believe φ,by taslım, i.e. ‘granting’ φ and deducing something clearlyfalse.

For Ibn Sına, talking about taslım is a way of talking aboutinference from assumptions (Ableitung), as opposed todemonstration that something is true (Schluss).

He doesn’t use it for RAA, presumably because inscientific proofs there simply is no opponent.

14

Ibn Sına also rejects explanations of RAA based oncounterfactual reasoning.

Reason: according to Ibn Sına, counterfactual inference isnon-monotonic.When we adopt a counterfactual assumption,we overrule some previous inferences by kicking out ofreach the premises on which they were based.In scientific discourse this is not allowed.

15

“It’s not true that if we stipulate that m = 2n then itfollows that m is even. One can stipulate animpossibility that prevents that. . . .But what we do [in the sciences] is to add acondition (sart.) which kicks out any conditions(surut.) that prevent the inference.For example ‘If m = 2n and nothing impossible isattributed to m, then m is even’.” (Qiyas 274.5–15)

16

Background on Ibn Sına’s view of language

For Ibn Sına, sentences have a basic structure and(usually) pieces added on.

He variously calls the added piecesI condition (sart. , plural surut.)I attachment (id. afa)I addition (ziyada)

17

Example (mine but based on many in Ibn Sına)

The sentence ‘He is eating’ contains no reference to thepresent. Proof: we can say ‘If a person chews andswallows, he is eating’. A reference to all times is implied.

So the sentence ‘He is eating’ has an unspokenattachment: ‘He is eating at time t’.

Then the longer sentence contains an unspoken condition:‘For all times t, if a person chews and swallows at time t,he is eating at time t’.Another unspoken condition: ‘If t is the present, he iseating at time t’.

18

(II) Ibn Sına’s evidence for the missing conditions

For Ibn Sına, a major part of logical analysis is to uncover(racay) additions which are implied but suppressed(mah. d

¯uf) in common usage. E.g.

“In common acceptance things are taken unquan-tified, while in the sciences they have to be takenquantified. When you pay attention to (racayta) thisin the examples above, . . . what is excellent in athing has to be excellent absolutely, i.e. excellentwithout any addition (ziyada).Being excellent in every respect . . . is being excellentwith an addition (ziyada), namely ‘in every respect’.”(Jadal 143.1–5)

19

Or from Ibn Sına’s Autobiography:

“The next year and a half I devoted myself entirelyto reading Philosophy: I read Logic and all theparts of philosophy once again. . . . I compiled a setof files for myself, and for each argument that Iexamined, I recorded the syllogistic premises itcontained, the way in which they were composed,and the conclusions which they might yield, andI would also pay attention to the conditions (’uracısurut.) of its premises,until I had checked out that particular problem.”

20

Ibn Sına claims that in common practice, the falseassumption θ in RAA is introduced by ‘If θ’, but ‘If θ’ isnot repeated when things are derived from θ.

We can check this from the Arabic of Euclid’s Elements 1(see the handout). Thus:

1. Ibn Sına is right that the false assumption is alwaysintroduced with ‘If’, not with ‘Let’ or ‘Suppose’.

2. In many cases the ‘If’ is immediately followed by a‘Suppose ζ’, where ζ includes everything in θ that will beused. Then ‘If ζ’ is not repeated, but we shouldn’t expectit to be. Ibn Sına never gives his views on ‘Suppose’.

21

3. In a significant number of cases ζ is missing or doesn’tcover everything used from θ. In these cases Ibn Sına isconfirmed; ‘If θ’ is assumed silently.

4. There is an example where ‘If θ’ is used silently,after an assumption that was not made for RAA.(In modern terms it was made for ∃E.)

5. There is also an example where during the derivation,θ is stated as a fact, not as a condition.If the condition ‘If θ’ was added, we would get the‘If θ then θ’ which appears in Ibn Sına’s account.

22

(III) How does Ibn Sına know that the derivationremains valid when the implied conditional clauses areadded?

There is no syllogistic rule that would guarantee it.My answer is a tad speculative,based partly on what Ibn Sına doesn’t say.

Note first that if χ consists of a proposition φ plusadditions (conditions etc.) then φ normally occurspositively in χ.For example ψ is a condition in (ψ → φ),but presumably not in (φ→ ψ).

23

Fact (in any standard natural deduction system):

Let Γ be a set of formulas and φ, ψ formulas. Let δ(p)be a formula in which p occurs only positively, andp is not in the scope of any quantifier on a variablefree in some formula of Γ. Suppose

Γ, φ ` ψ.

ThenΓ, δ(φ) ` δ(ψ).

Call this Ibn Sına’s Lemma. It gives exactly what he needsfor the preservation of validity.

24

Ibn Sına is almost obsessively interested in the differencebetween affirmative and negative, but he normally showsno awareness of the notion of a positive occurrence.

For example he has no notion of scope,and no glimmering of the laws of distribution.

Nevertheless his notion of ‘condition’ seems to havepositivity hidden in it.In those terms, Ibn Sına’s Lemma is a kind of statement ofupwards monotonicity.

25

For Ibn Sına, the logical properties of propositions aremainly the result of their basic structure.For example he has no examples of valid entailmentswhich depend on additions.

He writes as if, other things being equal,making additions in a uniform way to propositions in avalid syllogism leaves the syllogism valid.

For example modal syllogisms are predicative syllogismswith modalities added.His discussion of them is almost entirely about whatadditions of modalities will cancel the validity of theunderlying predicative syllogism.

26

Ibn Sına often talks about the mental processing ofsyllogisms.He emphasises that in predicative sentences,the analysis needed for logic never goes inside the subjectand the predicate.In this sense, logical rules apply only at the top syntacticlevel.

It was always a problem for traditional logicians to seehow we can make inferences that depend on deeperanalysis of the propositions. Burley, Buridan, Leibniz,De Morgan all worried at this problem.Burley had the idea of induction on complexity.The others relied on paraphrases,e.g. to bring the deeper structures to the top level.

27

It seems Ibn Sına has a solution:

Treat the upper layers of the proposition as‘additions’, and apply the logical rules directlyat the deeper level.

The solution is remarkably close to Frege’s.See for example Frege’s construction of Begriffsschrift sothat modus ponens can be applied directly to positivelyoccurring subformulas at any depth.Frege also has signs of Ibn Sına’s pretence that the uppersyntactic levels can be ignored.